Application of Yang homotopy perturbation transform approach for solving multi-dimensional diffusion problems with time-fractional derivatives

In this paper, we aim to present a powerful approach for the approximate results of multi-dimensional diffusion problems with time-fractional derivatives. The fractional order is considered in the view of the Caputo fractional derivative. In this analysis, we develop the idea of the Yang homotopy perturbation transform method (YHPTM), which is the combination of the Yang transform (YT) and the homotopy perturbation method (HPM). This robust scheme generates the solution in a series form that converges to the exact results after a few iterations. We show the graphical visuals in two-dimensional and three-dimensional to provide the accuracy of our developed scheme. Furthermore, we compute the graphical error to demonstrate the close-form analytical solution in the comparison of the exact solution. The obtained findings are promising and suitable for the solution of multi-dimensional diffusion problems with time-fractional derivatives. The main advantage is that our developed scheme does not require assumptions or restrictions on variables that ruin the actual problem. This scheme plays a significant role in finding the solution and overcoming the restriction of variables that may cause difficulty in modeling the problem.


Formulation of YHPTM
In this section, we construct the idea of YHPTM which is used to derive the approximate results of multidimensional diffusion problems with time-fractional derivatives.This scheme does not require the restriction of variables and any hypothesis.Let's assume the following differential problem of time-fractional order as with initial condition (2) D α τ ϑ(ℑ, τ ) = L 1 ϑ(ℑ, τ ) + L 2 ϑ(ℑ, τ ) + h(ℑ, τ ), www.nature.com/scientificreports/Operating YT on Eq. ( 2) such as

This implies
Hence R(ξ ) is evaluated such as Operating inverse YT on Eq. ( 4), it yields where Now, HPM is defined as and where H n polynomials are expressed as; Use Eqs. ( 6) and (7) in Eq. ( 5), it yields Comparing the coefficient of p, we obtain similarly, it can be continued to the following series Equation (9) represents the approximate solution of the fractional problem (2).

Convergence and error analysis
The following theorems are built on the idea of the proposed scheme and provided to show the convergence and error analysis of the problem (2) (3) ϑ(ℑ, 0) = k(ℑ).
This ends the proof.

Applications
We illustrate four applications of multi-dimensional diffusion problems with time-fractional derivatives.We consider two-dimensional and three-dimensional heat flow problems in the sense of Caputo fractional derivative.These examples exhibit the performance and capability of the presented scheme.Graphical results and absolute (10) . . . Vol

Example 1
Let us consider the two-dimensional homogeneous time-fractional heat flow problem with the initial condition Applying the YT on Eq. ( 15), we get The application of YT in fractional form yields Thus, R(ξ ) is obtained as Using inverse YT on Eq. ( 17), we get Implementing the idea of of HPM to derive the He's iterations Relating the similar components of p, we get Similarly, it can be continued to the following series which can be closed form In Fig. 1, we provide the graphical visuals of approximate series solution of Eq. ( 19) and the exact solution of Eq. ( 20) at −10 ≤ ℑ ≤ 10 and 0 ≤ τ ≤ 0.1 .These visuals indicate that when we increase the value of frac- tional order α , our graphical results approach to the exact graph significantly.We plotted the graphical error in two-dimensional visuals in Fig. 2 at α = 0.25, 0.50, 0.75, 1 .This shows comparison yields that YHPTM is fast and convenient approach.Table 1 presents the absolute errors between the approximate solution and the exact , , , solution of three-dimensional heat flow problem.This table shows that when α = 1 , our obtained values are very close to the exact solution than the values of α = 0.50 and the value of absolute error decreases precisely.

Example 2
Consider the following time-fractional heat flow problem in a inhomogeneous two-dimensional form with the initial condition Applying the YT on Eq. ( 21), we get The application of YT in fractional form yields Relating the similar components of p, we get  In Fig. 3, we provide the graphical visuals of approximate series solution of Eq. ( 25) and the exact solution of Eq. ( 26) at −1 ≤ ℑ ≤ 1 and 0 ≤ τ ≤ 0.5 .These visuals indicate that when we increase the value of fractional order α , our graphical results approach to the exact graph significantly.We plotted the graphical error in two- dimensional visuals in Fig. 4 at α = 0.25, 0.50, 0.75, 1 .This shows comparison yields that YHPTM is fast and convenient approach.Table 2 presents the absolute errors between the approximate solution and the exact solution of three-dimensional heat flow problem.This table shows that when α = 1 , our obtained values are very close to the exact solution than the values of α = 0.50 and the value of absolute error decreases precisely.

Example 3
Consider the following time-fractional heat flow problem in a three-dimensional homogeneous form with the initial condition Applying the YT on Eq. ( 27), we get Using the properties functions of YT , we obtain Thus R(ξ ) is obtained as Using inverse YT on Eq. ( 29), we get Implementing the idea of HPM to derive the He's iterations Relating the similar components of p, we get , , , , Similarly, it can be continued to the following series , , , In Fig. 5, we provide the graphical visuals of approximate series solution of Eq. ( 31) and the exact solution of Eq. (32) −3 ≤ ℑ ≤ 3 and 0 ≤ τ ≤ 0.1 .These visuals indicate that when we increase the value of fractional order α , our graphical results approach to the exact graph significantly.We plotted the graphical error in two- dimensional visuals in Fig. 6 at α = 0.25, 0.50, 0.75, 1 .This shows comparison yields that YHPTM is fast and convenient approach.Table 3 presents the absolute errors between the approximate solution and the exact solution of three-dimensional heat flow problem.This table shows that when α = 1 , our obtained values are very close to the exact solution than the values of α = 0.50 and the value of absolute error decreases precisely.

Example 4
Consider the following time-fractional heat flow problem in a three-dimensional inhomogeneous form with the initial condition The application of YT in fractional form yields (31) p 0 : ϑ 0 (ℑ, ℘, ̟ , τ ) = ϑ(ℑ, ℘, 0) = sin(ℑ + ℘) + sin ̟ + sin ̟ τ α Ŵ(α + 1) , , , , + sin ̟ τ 5α Ŵ(5α + 1) , . ...  In Fig. 7, we provide the graphical visuals of approximate series solution of Eq. ( 37) and the exact solution of Eq. (38) −1 ≤ ℑ ≤ 1 and 0 ≤ τ ≤ 0.5 .These visuals indicate that when we increase the value of fractional order α , our graphical results approach to the exact graph significantly.We plotted the graphical error in two- dimensional visuals in Fig. 8 at α = 0.25, 0.50, 0.75, 1 .This shows comparison yields that YHPTM is fast and convenient approach.Table 4 presents the absolute errors between the approximate solution and the exact solution of three-dimensional heat flow problem.This table shows that when α = 1 , our obtained values are very close to the exact solution than the values of α = 0.50 and the value of absolute error decreases precisely. (37)

Conclusion
In this study, we successfully developed the YHPTM approach for obtaining the approximate solution of the two-dimensional and three-dimensional heat flow problems.Since the equations involving fractional order are quite difficult to solve directly, we introduce the idea of YT to dissolve the fractional order of the problem.The scheme of YT is limited and unable to generate the series solution, therefore, we implement HPM to derive the successive iterations from the classical equation that leads the results to the exact solution very easily.We consider four test problems to show the efficiency and effectiveness of this proposed scheme.It has been found that our derived results demonstrate a great confirmation of compromise with the exact solution.We also analyzed the efficiency of our proposed scheme in two-dimensional and three-dimensional through graphical structures.The obtained results are efficient and significant, demonstrating that YHPTM is accurate and authentic for fractional problems.It is expected to consider this scheme for fractional problems in the sense of Atangana-Baleanu derivatives and other partial differential equations involving fractal theory and fractional calculus in our future work.

Table 1 .
Absolute error between the obtained results and the exact solution at ℘ = 0.5 and τ = 0.001.

Table 2 .
Absolute error between the obtained results and the exact solution at ℘ = 0.5 and τ = 0.005.
ℑ α = 0.50 YHPTM results at α = 1 Exact results Absolute error at α = 0.50 Absolute error at α = 1 Using the properties functions of YT, we obtain Thus R(ξ ) is obtained as Using inverse YT on Eq. (35), we get Implementing the idea of HPM to derive the He's iterations

Table 3 .
Absolute error between the obtained results and the exact solution at ℘ = ̟ = 0.5 and τ = 0.001.

Table 4 .
Absolute error between the obtained results and the exact solution at ℘ = ̟ = 0.5 and τ = 0.005.